Almost sure global nonlinear smoothing for the 2D NLS

TL;DR

This paper establishes an almost-sure global nonlinear smoothing effect for the 2D nonlinear Schrödinger equation on the torus, extending previous results from the circle to higher dimensions using Gaussian measure quasi-invariance.

Contribution

It proves a new almost-sure global smoothing phenomenon for 2D NLS on the torus, leveraging Gaussian measure quasi-invariance, which was previously unknown in higher dimensions.

Findings

Proves almost-sure global nonlinear smoothing for 2D NLS.

Extends smoothing results from circle to multidimensional torus.

Uses Gaussian measure quasi-invariance with covariance (1-Δ)^{-s} for s>2.

Abstract

In this article, we prove an almost-sure global in time nonlinear smoothing effect for NLS on the two-dimensional torus. For deterministic data, this phenomenon was proved for the NLS on the circle by Erdo\u{g}an--Tzirakis, which remains unknown on multidimensional torus. Our argument is based on a quantitative quasi-invariance of Gaussian measures with covariance operator $(1-\Delta)^{-s}$ for $s>2$.